Put the onions in a pot with a pinch of salt and part of the olive oil. Cook over medium-low heat, stirring occasionally, until the onions soften and become golden.
Add celery and cook 2 to 3 minutes. Add cabbage and cook 2 to 3 minutes. Add kale and cook 2 to 3 minutes, stirring from time to time.
Add the tomato and basil and stir well. Add the bouillon cube. Add enough water to cover the vegetables by about 2 inches. Add the cooked beans, cover, and simmer 30 minutes.
Take the pot off the heat. Stir in the remaining olive oil, the lemon juice, and a few grindings of pepper. Let stand about 10 minutes.
Preheat the oven to 400 F (205 C). Toast the bread slices. In a lidded ovenproof casserole, make 3 layers: bread, soup, and Parmesan. End with soup and cheese on top.
Break the eggs over the top. Spoon about 1 tbsp of soup over each egg. Cover and bake 6 minutes. Uncover and bake 4 minutes more. Serve with a wide spatula.